Explicit Interval Estimates for Prime Numbers

TL;DR

This paper develops explicit interval estimates for prime numbers using smoothing functions and Riemann zeta-function zeros, providing bounds on prime distribution within specific intervals for large x.

Contribution

It introduces a method to compute explicit prime intervals based on advanced analytic number theory techniques, improving understanding of prime distribution bounds.

Findings

Explicit bounds for prime intervals for large x

Method leveraging zeros of the Riemann zeta-function

Enhanced understanding of prime distribution patterns

Abstract

Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of $(\Delta,x_0)$ such that for all $x \geq x_0$ there exists at least one prime in the interval $(x(1 - \Delta^{-1}), x]$.